On Risk Model with Dividends Payments Perturbed by a Brownian Motion – An Algorithmic Approach

Assume that an insurance company pays dividends to its shareholders whenever the surplus process is above a given threshold. In this paper we study the expected amount of dividends paid, and the expected time to ruin in the compound Poisson risk process perturbed by a Brownian motion. Two models are considered: In the first one the insurance company pays whatever amount exceeds a given level b as dividends to its shareholders. In the second model, the company starts to pay dividends at a given rate, smaller than the premium rate, whenever the surplus up-crosses the level b. The dividends are paid until the surplus down-crosses the level a, a < b . We assume that the claim sizes are phase-type distributed. In the analysis we apply the multidimensional Wald martingale, and the multidimensional Asmussesn and Kella martingale.

[1]  E. Frostig.  On the expected time to ruin and the expected dividends when dividends are paid while the surplus is above a constant barrier , 2005 .

[2]  Hans U. Gerber,et al.  An introduction to mathematical risk theory , 1982 .

[3]  D. Derfla,et al.  On a modification of the classical risk process , 2007 .

[4]  Offer Kella,et al.  A multi-dimensional martingale for Markov additive processes and its applications , 2000, Advances in Applied Probability.

[5]  Jostein Paulsen,et al.  Optimal choice of dividend barriers for a risk process with stochastic return on investments , 1997 .

[6]  Ren Asmussen,et al.  Fitting Phase-type Distributions via the EM Algorithm , 1996 .

[7]  Olivier Deprez.  “Optimal Dividends: Analysis with Brownian Motion,” Ana C. Cebrián, Hans U. Gerber and Elias S.W. Shiu, January 2004 , 2004 .

[8]  Esther Frostig,et al.  The expected time to ruin in a risk process with constant barrier via martingales , 2005 .

[9]  Søren Asmussen,et al.  Matrix‐analytic Models and their Analysis , 2000 .

[10]  Asymptotic Theory for a Risk Process with a High Dividend Barrier , 2003 .

[11]  Hansjörg Albrecher,et al.  On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang(n) interclaim times , 2005 .

[12]  Xiaowen Zhou,et al.  On a Classical Risk Model with a Constant Dividend Barrier , 2005 .

[13]  Shuanming Li,et al.  The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion , 2006 .

[14]  Howard R. Waters,et al.  Some Optimal Dividends Problems , 2004, ASTIN Bulletin.

[15]  Zong Zhao-jun,et al.  ASYMPTOTIC THEORY FOR A RISK PROCESS WITH A HIGH DIVIDEND BARRIER , 2007 .

[16]  José Garrido,et al.  On a class of renewal risk models with a constant dividend barrier , 2004 .

[17]  Hans U. Gerber,et al.  On Optimal Dividend Strategies In The Compound Poisson Model , 2006 .

[18]  A Comparative Analysis Of Chronic And Nonchronic Insured Commercial Member Cost Trends , 2006 .

[19]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[20]  Robert L. Brown,et al.  Further Analysis of Future Canadian Health Care Costs , 2004 .

[21]  Gordon E. Willmot,et al.  The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function , 2003 .

[22]  X. Sheldon Lin,et al.  The compound Poisson risk model with a threshold dividend strategy , 2006 .

[23]  Søren Asmussen,et al.  Ruin probabilities , 2001, Advanced series on statistical science and applied probability.

[24]  E. Mordecki.  The distribution of the maximum of a Lévy process with positive jumps of phase-type ∗ , 2022 .

[25]  H. Gerber,et al.  On the Time Value of Ruin , 1997 .

[26]  Søren Asmussen,et al.  Computational methods in risk theory: A matrix-algorithmic approach , 1992 .

[27]  Onno Boxma,et al.  Useful Martingales for Stochastic Storage Processes with Lévy-Type Input , 2013, Journal of Applied Probability.